Misere combinatorial games.
This is [Davies, Miller, Milley (Definition 3.4 on p. 9)][davies:SumsPFreeForms:2025].
- misereOutcome_of_add_LL {g h : G} (h1 : PFreeSubset A g) (h2 : PFreeSubset A h) (h3 : Form.Misere.Outcome.MisereOutcome g = Outcome.L) (h4 : Form.Misere.Outcome.MisereOutcome h = Outcome.L) : Form.Misere.Outcome.MisereOutcome (g + h) = Outcome.L
- misereOutcome_of_add_RR {g h : G} (h1 : PFreeSubset A g) (h2 : PFreeSubset A h) (h3 : Form.Misere.Outcome.MisereOutcome g = Outcome.R) (h4 : Form.Misere.Outcome.MisereOutcome h = Outcome.R) : Form.Misere.Outcome.MisereOutcome (g + h) = Outcome.R
- miserePlayerOutcome_of_add_LN {g h : G} (h1 : PFreeSubset A g) (h2 : PFreeSubset A h) (h3 : Form.Misere.Outcome.MisereOutcome g = Outcome.L) (h4 : Form.Misere.Outcome.MisereOutcome h = Outcome.N) : Form.Misere.Outcome.MiserePlayerOutcome (g + h) Player.left = Player.left
- miserePlayerOutcome_of_add_RN {g h : G} (h1 : PFreeSubset A g) (h2 : PFreeSubset A h) (h3 : Form.Misere.Outcome.MisereOutcome g = Outcome.R) (h4 : Form.Misere.Outcome.MisereOutcome h = Outcome.N) : Form.Misere.Outcome.MiserePlayerOutcome (g + h) Player.right = Player.right
Instances
If $\mathcal{A}$ is outcome-stable, $n \in \mathbb{N}$ then $0 \ge_{\operatorname{pf}(\mathcal{A})} 1$.
If $\mathcal{A}$ is outcome-stable, $n \in \mathbb{N}$ then $n \ge_{\operatorname{pf}(\mathcal{A})} 1 + n$.
If $\mathcal{A}$ is outcome-stable, $n, m \in \mathbb{N}$, and $n \le m$ then $n \ge_{\operatorname{pf}(\mathcal{A})} m$.
If $\mathcal{A}$ is outcome-stable, $n \in \mathbb{Z}$ then $n \ge_{\operatorname{pf}(\mathcal{A})} 1 + n$.
If $\mathcal{A}$ is outcome-stable, $n \in \mathbb{N}$, $k \in \mathbb{Z}$ then $k \ge_{\operatorname{pf}(\mathcal{A})} n + k$.
If $\mathcal{A}$ is outcome-stable, $n, m \in \mathbb{Z}$, and $n \le m$ then $n \ge_{\operatorname{pf}(\mathcal{A})} m$.
If $\mathcal{A}$ is outcome-stable and $G \in \operatorname{pf}(\mathcal{A})$ then $\operatorname{o}(G + 1) \le \operatorname{o}(G)$.
If $\mathcal{A}$ is outcome-stable and $G \in \operatorname{pf}(\mathcal{A})$ then $\operatorname{o}(G) \le \operatorname{o}(G - 1)$.
If $\mathcal{A}$ is outcome-stable, $G \in \operatorname{pf}(\mathcal{A})$, and $n \in \mathbb{N}$ then $\operatorname{o}(G + n) \le \operatorname{o}(G)$.
If $\mathcal{A}$ is outcome-stable, $G \in \operatorname{pf}(\mathcal{A})$, and $k, m \in \mathbb{Z}$ and $k \le m$ then $\operatorname{o}(G + m) \le \operatorname{o}(G + k)$.
If $G$ is $\mathscr{P}$-free and $n$ is an integer then $G + n$ is also $\mathscr{P}$-free.
This is [Davies, Miller, Milley (Lemma 3.3 on p. 9)][davies:SumsPFreeForms:2025].
The $\mathscr{L}$-region is downward closed: if $\operatorname{o}(G + m) = \mathscr{L}$ and $k \le m$, then $\operatorname{o}(G + k) = \mathscr{L}$.
The $\mathscr{R}$-region is upward closed: if $\operatorname{o}(G + k) = \mathscr{R}$ and $k \le m$, then $\operatorname{o}(G + m) = \mathscr{R}$.
The outcome of an integer shift is always one of $\mathscr{L}$, $\mathscr{N}$, $\mathscr{R}$ (never $\mathscr{P}$).
The $\mathscr{N}$-tipping point of a Left-win game lies strictly above $0$, and its witness is the positive shift: $\operatorname{o}(G + \operatorname{n}(G)) = \mathscr{N}$.
Positive $\mathscr{N}$-region for a next-win game: if $\operatorname{o}(G) = \mathscr{N}$ and $k < \operatorname{r}(G)$, then $\operatorname{o}(G + k) = \mathscr{N}$.
Negative $\mathscr{N}$-region for a next-win game: if $\operatorname{o}(G) = \mathscr{N}$ and $k < \operatorname{l}(G)$, then $\operatorname{o}(G - k) = \mathscr{N}$.
For a Left-win game, the $\mathscr{N}$-tipping point lies strictly below the $\mathscr{R}$-tipping point.
Shifting by a natural translates the $\mathscr{R}$-tipping point: $\operatorname{r}(G + k) = \operatorname{r}(G) - k$.
For a Left-win game, shifting by a natural $k \le \operatorname{n}(G)$ translates the $\mathscr{N}$-tipping point: $\operatorname{n}(G + k) = \operatorname{n}(G) - k$.
If $\mathcal{A}$ is outcome-stable, hereditary monoid containing $1$ and $-1$, and $G \in \operatorname{pf}(\mathcal{A})$, then $\operatorname{n}(G^L) \le \operatorname{r}(G)$ for all Left options $G^L$ of $G$ with $\operatorname{o}(G^L) \ne \mathscr{R}$.
This is [Davies, Miller, Milley (Lemma 3.7 on p. 13)][davies:SumsPFreeForms:2025].
If $\mathcal{A}$ is outcome-stable, hereditary monoid containing $1$ and $-1$, and $G \in \operatorname{pf}(\mathcal{A})$, then $\operatorname{n}(G^R) \le \operatorname{l}(G)$ for all Right options $G^R$ of $G$ with $\operatorname{o}(G^R) \ne \mathscr{L}$.
This is the mirror of [Davies, Miller, Milley (Lemma 3.7 on p. 13)][davies:SumsPFreeForms:2025].
If $G$ is a $\mathscr{P}$-free Left end with $\operatorname{o}(G) = \mathscr{N}$, then $\operatorname{r}(G) = 1$.
This is [Davies, Miller, Milley (Lemma 3.8 on p. 13)][davies:SumsPFreeForms:2025].
If $\mathcal{A}$ is outcome-stable, hereditary monoid containing $1$ and $-1$, and $G \in \operatorname{pf}(\mathcal{A})$ with $\operatorname{o}(G) = \mathscr{N}$, then either $G$ is Left end-like with $\operatorname{r}(G) = 1$, or else there exists a Left option $G^L$ of $G$ with $\operatorname{o}(G^L) = \mathscr{L}$ such that $\operatorname{n}(G^L) = r(G)$.
This is [Davies, Miller, Milley (Lemma 3.9 on p. 13)][davies:SumsPFreeForms:2025].
If $\mathcal{A}$ is outcome-stable, hereditary monoid containing $1$ and $-1$, and $G \in \operatorname{pf}(\mathcal{A})$ with $\operatorname{o}(G) = \mathscr{N}$, then either $G$ is Right end-like with $\operatorname{l}(G) = 1$, or else there exists a Left option $G^R$ of $G$ with $\operatorname{o}(G^R) = \mathscr{R}$ such that $\operatorname{n}(G^R) = l(G)$.
This is mirror of [Davies, Miller, Milley (Lemma 3.9 on p. 13)][davies:SumsPFreeForms:2025].
If $\mathcal{A}$ is outcome-stable, hereditary monoid containing $1$ and $-1$, and $G \in \operatorname{pf}(\mathcal{A})$ is a Left end then $\operatorname{r}(G) = \operatorname{n}(G) + 1$.
This is [Davies, Miller, Milley (Lemma 3.10 on p. 13)][davies:SumsPFreeForms:2025].
If $\mathcal{A}$ is outcome-stable, hereditary monoid containing $1$ and $-1$, and $G \in \operatorname{pf}(\mathcal{A})$ is a Right end then $\operatorname{l}(G) = \operatorname{n}(G) + 1$.
This is the mirror of [Davies, Miller, Milley (Lemma 3.10 on p. 13)][davies:SumsPFreeForms:2025].
If $\mathcal{A}$ is outcome-stable, hereditary, integer-invertible monoid, and $G \in \operatorname{pf}(\mathcal{A})$ with $\operatorname{o}(G) = \mathscr{L}$ and $G^R$ is a Right option of $G$, then $\operatorname{r}(G^R) \ge \operatorname{n}(G)$.
If $\mathcal{A}$ is outcome-stable, hereditary, integer-invertible monoid, and $G \in \operatorname{pf}(\mathcal{A})$ with $\operatorname{o}(G) = \mathscr{L}$ and $G^L$ is a Left option of $G$, then $\operatorname{l}(G^L) \ge \operatorname{n}(G)$.
If $\mathcal{A}$ is outcome-stable, hereditary, integer-invertible monoid, and $G \in \operatorname{pf}(\mathcal{A})$ with $\operatorname{o}(G) = \mathscr{L}$ then there exists a Right option $G^R$ of $G$ with $\operatorname{r}(G^R) = \operatorname{n}(G)$.
If $\mathcal{A}$ is outcome-stable, hereditary, integer-invertible monoid, and $G \in \operatorname{pf}(\mathcal{A})$ with $\operatorname{o}(G) = \mathscr{R}$ then there exists a Left option $G^L$ of $G$ with $\operatorname{l}(G^L) = \operatorname{n}(G)$.
If $\mathcal{A}$ is outcome-stable, hereditary monoid containing $1$ and $-1$, and $G \in \operatorname{pf}(\mathcal{A})$ with $\operatorname{o}(G) = \mathscr{L}$, then if $\operatorname{n}(G) \ne \operatorname{r}(G) - 1$ then there exists some option $G^L$ with $\operatorname{o}(G^L) = \mathscr{L}$ such that $\operatorname{n}(G^L) = \operatorname{r}(G)$.
This is (1) in [Davies, Miller, Milley (Lemma 3.11 on p. 14)][davies:SumsPFreeForms:2025].
If $\mathcal{A}$ is outcome-stable, hereditary monoid containing $1$ and $-1$, and $G \in \operatorname{pf}(\mathcal{A})$ with $\operatorname{o}(G) = \mathscr{R}$, then if $\operatorname{n}(G) \ne \operatorname{l}(G) - 1$ then there exists some option $G^R$ with $\operatorname{o}(G^R) = \mathscr{R}$ such that $\operatorname{n}(G^R) = \operatorname{l}(G)$.
This is mirror of (1) in [Davies, Miller, Milley (Lemma 3.11 on p. 14)][davies:SumsPFreeForms:2025].